Preheat your oven to 350°F (175°C).
In a medium saucepan, melt the butter over low heat. Once melted, remove from heat and let it cool until warm.
In a large mixing bowl, combine the melted butter, light brown sugar, and granulated sugar. Whisk together until smooth.
Add the large egg, egg yolk, and vanilla extract to the sugar mixture. Whisk until fully incorporated.
In another bowl, whisk together the all-purpose flour, baking soda, and salt. Gradually add this dry mixture to the wet ingredients, stirring gently until just combined.
Gently fold in the semi-sweet and milk chocolate chips until evenly distributed.
Pour the cookie dough into your preheated cast-iron skillet, spreading it evenly.
Bake in the preheated oven for 25-30 minutes, or until the edges are golden brown and the center is set but slightly soft.
Allow the skillet cookie to cool for about 10 minutes before slicing. Serve warm, optionally topped with ice cream or whipped cream.
